JHU Number Theory Seminar, Spring 2005

Abstract of Talks

Febuary 9, 4:30pm--5:30pm

Kazuya Kato, Ramification theory for schemes
This is a joint work with Takeshi Saito. In this talk, we generalize the classical ramification theory of discrete valuations (differents, conductors, etc) to the ramification theory for higher dimensional schemes. Professor Ono asks me to show my "dance of prime numbers". I am afraid you will be shocked too much if I show such a strange thing.

Febuary 16, 4:30pm--5:30pm

Wenzhi Luo, Equidistribution and quantum variance for Hecke eigenforms
In this talk, I'd like to describe my recent work with P.Sarnak concerning the quantum variance on the modular surface for Hecke eigenforms. We compute the quantum variance using the trace formula and analyze its subtle arithmetic structure and connections to Hecke operators and central values of triple product L-functions.

March 9, 4:30pm--5:30pm

Zhengyu Mao, Shimura correspondence for newforms
Kohnen-Zagier formula is one of many versions of a formula relating the central value of twisted L-functions L(f,D,k) of a weight 2k cusp form and the |D|-th Fourier coefficient of a half integral weight form. We describe a formula that holds for any newform of odd level, and any fundamental discriminant D. For this generalization we need to define Shimura correspondence in a less strict sense.

March 30, 4:30pm--5:30pm

Morten Risager, Densities of homology classes
I will dicuss some very recent equidistribution results about densities of geodesics whose homology class lies in a big fixed set. The main tool is character perturbations of Laplacian. This is a joint work with Y. Petridis.

April 13, 4:30pm--5:30pm

Matthew Boylan, Non-vanishing of central critical values of modular L-functions modulo p
Let F(z) be a newform of weight 2k, let D be a fundamental discriminant, and let L(F,D,s) be the L-series of F twisted by the Kronecker character of discriminant D. In this talk, I will show that if there are two D (subject to some local conditions) for which the algebraic part of the central critical value L(F,D,k) is not 0 (mod p), then there are infinitely many such D. This result depends on non-vanishing results for the Fourier coefficients of half-integral weight modular forms modulo p, which are of independent interest. I will also discuss applications to elliptic curves and orders of Shafarevich-Tate groups.